BOOK I.

OF THE MOTION OF BODIES.

SECTION III.

If a body revolves in an ellipsis; it is required to find the law of
the centripetal force tending to the focus of the ellipsis.

Let S be the focus of the ellipsis. Draw SP cutting the diameter DK of the ellipsis in
E, and the ordinate Qv in x; and complete the parallelogram QxPR. It
is evident that EP is equal to the greater semi-axis AC: for drawing HI from the other
focus H of the ellipsis parallel to EC, because CS, CH are equal, ES, EI will be also
equal; so that EP is the half sum of PS, PI, that is (because of the parallels HI, PR, and
the equal angles IPR, HPZ), of PS, PH, which taken together, are equal to the whole axis
2AC. Draw QT perpendicular to SP, and putting L for the principal latus rectum of the
ellipsis (or for 2BC2/AC), we shall have L · QR to L · Pv as QR to Pv,
that is, as PE or AC to PC; and L · Pv to GvP as L to Gv; and GvP
to Qv2 as PC2 to CD2; and by (Corol. 2, Lem. VII) the points Q and P coinciding, Qv2
is to Qx2 in the ratio of equality; and Qx2 or Qv2
is to QT2 as EP2 to PF2, that is, as CA2 to PF2,
or (by Lem. XII) as CD2 to CB2. And
compounding all those ratios together, we shall have L · QR to QT2 as AC · L
· PC2 · CD2, or 2CB2 · PC2 · CD2
to PC · Gv · CD2 · CB2, or as, 2PC to Gv. But the
points Q and P coinciding, 2PC and Gr are equal. And therefore the quantities L ·
QR and QT2, proportional to these, will be also equal. Let those equals be
drawn into SP2/QR, and L · SP2 will become equal to SP2·QT2 / QR. And therefore (by Corol. 1
and 5, Prop. VI) the centripetal force is
reciprocally as L · SP2, that is, reciprocally in the duplicate ratio of the
distance SP. Q.E.D.

The same otherwise.

Since the force tending to the centre of the ellipsis, by which the body P may revolve
in that ellipsis, is (by Corol. 1, Prop. X.) as the
distance CP of the body from the centre C of the ellipsis; let CE be drawn parallel to the
tangent PR of the ellipsis; and the force by which the same body P may revolve about any
other point S of the ellipsis, if CE and PS intersect in E, will be as PE2/SP2
(by Cor. 3, Prop. VII.); that is, if the point S
is the focus of the ellipsis, and therefore PE be given as SP2 reciprocally.
Q.E.D.

With the same brevity with which we reduced the fifth
Problem to the parabola, and hyperbola, we might do like here: but because of the
dignity of the Problem and its use in what follows, I shall confirm the other cases by
particular demonstrations.

Suppose a body to move in an hyperbola; it is required to find the
law of the centripetal force tending to the focus of that figure.

Let CA, CB be the semi-axes of the hyperbola; PG, KD other conjugate diameters; PF a
perpendicular to the diameter KD; and Qv an ordinate to the diameter GP. Draw SP
cutting the diameter DK in E, and the ordinate Qv in x, and complete the
parallelogram QRPx. It is evident that EP is equal to the semi-transverse axis AC;
for drawing HI, from the other focus H of the hyperbola, parallel to EC, because CS, CH
are equal, ES, EI will be also equal; so that EP is the half difference of PS, PI; that is
(because of the parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the
difference of which is equal to the whole axis 2AC. Draw QT perpendicular to SP; and
putting L for the principal latus rectum of the hyperbola (that is, for 2BC2/AC),
we shall have L · QR to L · Pv as QR to Pv, or Px to Pv,
that is (because of the similar triangles Pxv, PEC), as PE to PC, or AC to PC. And
L · Pv will be to Gv · Pv, as L to Gv; and (by the
properties of the conic sections) the rectangle GvP is to Qv2 as
K, PC2 to CD2; and by (Cor. 2,
Lem. VII.), Qv2 to Qx2, the points Q and P
coinciding, becomes a ratio of equality; and Qx2 or Qv2
is to QT2 as EP2 to PF2, that is, as CA2 to PF2,
or (by Lem. XII.) as CD2 to CB2:
and, compounding all those ratios together, we shall have L · QR to QT2 as AC
· L · PC2 · CD2, or 2CB2 · PC2 · CD2
to PC · Gv · CD2 · CB2, or as 2PC to Gv. But the
points P and Q coinciding, 2PC and Gv are equal. And therefore the quantities L ·
QR and QT2, proportional to them, will be also equal. Let those equals be drawn
into SP2/QR, and we shall have L · SP2 equal to SP2· QT2
/ QR. And therefore (by Cor. 1 and 5, Prop. VI.) the centripetal force is reciprocally as
L · SP2, that is, reciprocally in the duplicate ratio of the distance SP.
Q.E.D.

The same otherwise.

Find out the force tending from the centre C of the hyperbola. This will be
proportional to the distance CP. But from thence (by Cor.
3, Prop. VII.) the force tending to the focus S will be as PE3/SP2
that is, because PE is given reciprocally as SP2. Q.E.D.

And the same way may it be demonstrated, that the body having its centripetal changed
into a centrifugal force, will move in the conjugate hyperbola.

The perpendicular, let fall from the focus of a parabola on its
tangent, is a mean proportional between the distances of the focus from the point of
contact, and from the principal vertex of the figure.

For, let AP be the parabola, S its focus, A its principal vertex, P the point of
contact, PO an ordinate to the principal diameter, PM the tangent meeting the principal
diameter in M, and SN the perpendicular from the focus on the tangent: join AN, and
because of the equal lines MS and SP, MN and NP, MA and AO, the right lines AN, OP, will
be parallel; and thence the triangle SAN will be right-angled at A, and similar to the
equal triangles SNM, SNP; therefore PS is to SN as SN to SA. Q.E.D.

If a body moves in the perimeter of a parabola; it is required to
find the law of the centripeta1 force tending to the focus of that figure.

Retaining the construction of the preceding Lemma, let P be the body in the perimeter
of the parabola; and from the place Q, into which it is next to succeed, draw QR parallel
and QT perpendicular to SP, as also Qv parallel to the tangent, and meeting the
diameter PG in v, and the distance SP in x. Now, because of the similar
triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or
QR and Pv of the other will be also equal. But (by the conic sections) the square
of the ordinate Qv, is equal to the rectangle under the latus rectum and the
segment Pv of the diameter; that is (by Lem. XIII.),
to the rectangle 4PS · Pv, or 4PS · QR; and the points P and Q coinciding, the
ratio of Qv to Qx, (by Cor. 2, Lem. VII.,)
becomes a ratio of equality. And therefore Qx2, in this case, becomes
equal to the rectangle 4PS · QR. But (because of the similar triangles QxT, SPN),
Qx2 is to QT2 as PS2 to SN2, that is
(by Cor. 1, Lem. XIV.), as PS to SA; that is, as
4PS · QR to 4SA · QR, and therefore (by Prop. IX. Lib. V., Elem.) QT2 and 4SA · QR are
equal. Multiply these equals by SP2/QR, and SP2·QT2 / QR
will become equal to SP2 · 4SA: and therefore (by Cor. 1 and 5, Prop.
VI.), the centripetal force is reciprocally as SP2 · 4SA; that is, because
4SA is given, reciprocally in the duplicate ratio of the distance SP. Q.E.D.

Cor. 1. From the three last Propositions it follows, that
if any body P goes from the place P with any velocity in the direction of any right line
PR, and at the same time is urged by the action of a centripetal force that is
reciprocally proportional to the square of the distance of the places from the centre, the
body will move in one of the conic sections, having its focus in the centre of force; and
the contrary. For the focus, the point of contact, and the position of the tangent, being
given, a conic section may be described, which at that point shall have a given curvature.
But the curvature is given from the centripetal force and velocity of the body being
given; and two orbits, mutually touching one the other, cannot be described by the same
centripetal force and the same velocity.

Cor. 2. If the velocity with which the body goes from its
place P is such, that in any infinitely small moment of time the lineola PR may be thereby
described; and the centripetal force such as in the same time to move the same body
through the space QR; the body will move in one of the conic sections, whose principal
latus rectum is the quantity QT2/QR in its ultimate state when the lineolæ PR,
QR are diminished in infinitum. In these Corollaries I consider the circle as an
ellipsis; and I except the case where the body descends to the centre in a right line.

If several bodies revolve about one common centre, and the
centripetal force is reciprocally in the duplicate ratio of the distance of places from
the centre; I say, that the principal latera recta of their orbits are in the duplicate
ratio of the areas, which the bodies by radii drawn to the centre describe in the same
time.

For (by Cor 2, Prop. XIII) the latus rectum L
is equal to the quantity QT2/QR in its ultimate state when the points P and Q
coincide. But the lineola QR in a given time is as the generating centripetal force; that
is (by supposition), reciprocally as SP2. And therefore QT2/QR is as
QT2 · SP2; that is, the latus rectum L is in the duplicate ratio of
the area QT · SP. Q.E.D.

Cor. Hence the whole area of the ellipsis, and the rectangle
under the areas, which is proportional to it, is in the ratio compounded of the
subduplicate ratio of the latus rectum, and the ratio of the periodic time. For the whole
area is as the area QT · SP, described in a given time, multiplied by the periodic time.

The same things being supposed, I say, that the periodic times in
ellipses are in the sesquiplicate ratio of their greater axes.

For the lesser axis is a mean proportional between the greater axis and the latus
rectum; and, therefore, the rectangle under, the areas is in the ratio compounded of the
subduplicate ratio of the latus rectum and the sesquiplicate ratio of the greater axis.
But this rectangle (by Cor. 3, Prop. XIV) is in a
ratio compounded of the subduplicate ratio of the latus rectum, and the ratio of the
periodic time. Subduct from both sides the subduplicate ratio of the latus rectum, and
there will remain the sesquiplicate ratio of the greater axis, equal to the ratio of the
periodic time. Q.E.D.

Cor. Therefore the periodic times in ellipsis are the same as
in circles whose diameters are equal to the greater axes of the ellipses.

The same things being supposed, and right lines being drawn to the
bodies that shall touch the orbits, and perpendiculars being let fall on those tangents
form the common focus; I say, that the velocities of the bodies are in a ratio compounded
of the ratio of the perpendiculars inversely, and the subduplicate ratio of the principal
latera recta directly.

From the focus S draw SY perpendicular to the tangent PR, and the velocity of the body
P will be reciprocally in the subduplicate ratio. Of the quantity SY2/L. For
that velocity is as the infinitely small arc PQ described in a given moment of time, that
is (by Lem. VII), as the tangent PR; that is (because of
the proportionals PR to QT, and SP to SY), as SP· QT / SY; or as SY reciprocally, and SP
· QT directly; but SP · QT is as the area described in the given time, that is (by Prop. XIV), in the subduplicate ratio of the latus rectum.
Q.E.D.

Cor. 1. The principal latera recta are in a ratio
compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the
velocities.

Cor. 2. The velocities of bodies, in their greatest and
least distances from the common focus, are in the ratio compounded of the ratio of the
distances inversely, and the subduplicate ratio of the principal latera recta directly.
For those perpendiculars are now the distances.

Cor. 3. And therefore the velocity in a conic section, at
its greatest or least distance from the focus, is to the velocity in a circle, at the same
distance from the centre, in the subduplicate ratio of the principal latus rectum to the
double of that distance.

Cor. 4. The velocities of the bodies revolving in
ellipses, at their mean distances from the common focus, are the same as those of bodies
revolving in circles, at the same distances; that is (by Cor. 6, Prop. IV), reciprocally in the subduplicate
ratio of the distances. For the perpendiculars are now the lesser semi-axes, and these are
as mean proportionals between the distances and the latera recta. Let this ratio inversely
be compounded with the subduplicate ratio of the latera recta directly, and we shall have
the subduplicate ratio of the distance inversely.

Cor. 5. In the same figure, or even in different figures,
whose principal latera recta are equal, the velocity of a body is reciprocally as the
perpendicular let fall from the focus on the tangent.

Cor. 6. In a parabola, the velocity is reciprocally in the
subduplicate ratio of the distance of the body from the focus of the figure; it is more
variable in the ellipsis, and less in the hyperbola, than according to this ratio. For (by
Cor. 2, Lem. XIV) the perpendicular let fall from
the focus on the tangent of a parabola is in the subduplicate ratio of the distance. In
the hyperbola the perpendicular is less variable; in the ellipsis more.

Cor. 7. In a parabola, the velocity of a body at any
distance from the focus is to the velocity of a body revolving in a circle, at the same
distance from the centre, in the subduplicate ratio of the number 2 to 1; in the ellipsis
it is less, and in the hyperbola greater, than according to this ratio. For (by Cor. 2 of this Prop.) the velocity at the vertex of a
parabola is in this ratio, and (by Cor. 6 of this
Prop. and Prop. IV) the same proportion holds in all
distances. And hence, also, in a parabola, the velocity is everywhere equal to the
velocity of a body revolving in a circle at half the distance; in the ellipsis it is less,
and in the hyperbola greater.

Cor. 8. The velocity of a body revolving in any section is
to the velocity of a body revolving in a circle, at the distance of half the principal
latus rectum of the section, as that distance to the perpendicular let fall from the focus
on the tangent of the section. This appears from Cor. 5.

Cor. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body revolving in
this circle is to the velocity of another body revolving in any other circle reciprocally
in the subduplicate ratio of the distances; therefore, ex æquo, the velocity of a
body revolving in a conic section will be to the velocity of a body revolving in a circle
at the same distance as a mean proportional between that common distance, and half the
principal latus rectum of the section, to the perpendicular let fall from the common focus
upon the tangent of the section.

Supposing the centripetal force to be reciprocally proportional to
the squares of the distances of places from the centre, and that the absolute quantity of
that force is known; it is required to determine the line which a body will describe that
is let go from a given place with a given velocity in the direction of a given right line.

Let the centripetal force tending to the point S be such as will make the body p
revolve in any given orbit pq; and suppose the velocity of this body in the place p
is known. Then from the place P suppose the body P to be let go with a given velocity
in the direction of the line PR; but by virtue of a centripetal force to be immediately
turned aside from that right line into the conic section PQ. This, the right line PR will
therefore touch in P. Suppose likewise that the right line pr touches the orbit pq
in p; and if from S you suppose perpendiculars let fall on those tangents, the
principal latus rectum of the conic section (by Cor.
1, Prop. XVI) will be to the principal latus rectum of that orbit in a ratio
compounded of the duplicate ratio of the perpendiculars, and the duplicate ratio of the
velocities; and is therefore given. Let this latus rectum be L; the focus S of the conic
section is also given. Let the angle RPH be the complement of the angle RPS to two right;
and the line PH, in which the other focus H is placed, is given by position. Let fall SK
perpendicular on PH, and erect the conjugate semi-axis BC; this done, we shall have SP2
- 2KPH + PH2 = SH2 = 4CH2 = 4BH2 - 4BC2
= SP + PH2 - L · SP + PH = SP2 + 2SPH + PH2
- L · SP + PH. Add on both sides 2KPH
 SP2 - PH2 + L · SP +
PH, and we shall have L · SP + PH =
2SPH + 2KPH, or SP + PH, to PH, as 2SP + 2KP to L. Whence PH is given both in length and
position. That is, if the velocity of the body in P is such that the latus rectum L is
less than 2SP + 2KP, PH will lie on the same side of the tangent PR with the line SP; and
therefore the figure will be an ellipsis, which from the given foci S, H, and the
principal axis SP + PH, is given also. But if the velocity of the body is so great, that
the latus rectum L becomes equal to 2SP + 2KP, the length PH will be infinite; and
therefore, the figure will be a parabola, which has its axis SH parallel to the line PK,
and is thence given. But if the body goes from its place P with a yet greater velocity,
the length PH is to be taken on the other side the tangent; and so the tangent passing
between the foci, the figure will be an hyperbola having its principal axis equal to the
difference of the lines SP and PH, and thence is given. For if the body, in these cases,
revolves in a conic section so found, it is demonstrated in Prop. XI, XII, and XIII, that
the centripetal force will be reciprocally as the square of the distance of the body from
the centre of force S; and therefore we have rightly determined the line PQ, which a body
let go from a given place P with a given velocity, and in the direction of the right line
PR given by position, would describe with such a force. Q.E.D.

Cor. 1. Hence in every conic section, from the principal
vertex D, the latus rectum L, and the focus S given, the other focus H is given, by taking
DH to DS as the latus rectum to the difference between the latus rectum and 4DS. For the
proportion, SP + PH to PH as 2SP + 2KP to L, becomes, in the case of this Corollary, DS +
DH to DH as 4DS to L, and by division DS to DH as 4DS - L to L.

Cor. 2. Whence if the velocity of a body in the principal
vertex D is given, the orbit may be readily found; to wit, by taking its latus rectum to
twice the distance DS, in the duplicate ratio of this given velocity to the velocity of a
body revolving in a circle at the distance DS (by Cor.
3, Prop. XVI.), and then taking DH to DS as the latus rectum to the difference between
the latus rectum and 4DS.

Cor. 3. Hence also if a body move in any conic section
and is forced out of its orbit by any impulse, you may discover the orbit in which it will
afterwards pursue its course. For by compounding the proper motion of the body with that
motion, which the impulse alone would generate, you will have the motion with which the
body will go off from a given place of impulse in the direction of a right line given in
position.

Cor. 4. And if that body is continually disturbed by the
action of some foreign force, we may nearly know its course, by collecting the changes
which that force introduces in some points, and estimating the continual changes it will
undergo in the intermediate places, from the analogy that appears in the progress of the
series.

If a body P, by means of a centripetal force tending to any given point R, move in the
perimeter of any given conic section whose centre is C; and the law of the centripetal
force is required: draw CG parallel to the radius RP, and meeting the tangent PG of the
orbit in G; and the force required (by Cor. 1, and Schol. Prop. X., and Cor. 3, Prop. VII.) will be as CG3/RP2.